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On operators with an absolute value condition. (English) Zbl 1076.47015

Let \(A\) be a bounded linear operator on a complex Hilbert space \(H\). An operator \(A\) is said to be a class A operator if and only if \(| A| ^{2}\leq | A^{2}| \) holds. An operator \(A\) is said to be a polynomially class A operator if \(p(A)\) is in class A with a nontrivial polynomial \(p\). It is well-known that class A contains the class of \(p\)-hyponormal operators for \(p>0\).
The authors show that (i) class A operators are finitely ascensive, (ii) \(A \otimes B\) belongs to class A if and only if \(A\) and \(B\) belong to class A, and (iii) Weyl’s theorem holds for \(f(A)\) if \(A\) is a polynomially class A operator and \(f\) is an analytic \(H\)-valued function on a neighborhood of \(\sigma(A)\).
Result (ii) is an extension of the second author’s [Glasg. Math. J. 42, 371–381 (2000; Zbl 0990.47017)] in which the case that \(A\) is \(p\)-hyponormal is treated. Result (iii) completely extends [B. P. Duggal and S. V. Djordjević, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 40, 49–56 (2000; Zbl 0997.47019)] and [Y. M. Hand and W. Y. Lee, Proc. Am. Math. Soc. 128, 2291–2296 (2000; Zbl 0953.47018)] through slightly different approaches.

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A80 Tensor products of linear operators
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